Denote by \(\mathcal {F}_k\)  the filtration generated by \(\lbrace  w_1, \dots , w_k\rbrace \) . It holds that \(\mathbb {E}[\widehat{\nabla } \ell (w_{k})|\mathcal {F}_k] = \nabla \ell (w_k)\) . Furthermore, almost surely, \(\mathbb {E}[\Vert \eta _{k+1}\Vert ^2|\mathcal {F}_k] \le 2 \mathbb {E}[\Vert \widehat{\nabla } \ell (w_{k})\Vert ^2| \mathcal {F}_k] +2 \Vert \nabla \ell (w_k)\Vert ^2 < 4 M_{\ell } \) , where \(M_{\ell } = \sup _{ 1\le j \le N}M_{\ell _j}\) .
Thus, for \(i \in \lbrace 1, \dots , d\rbrace \) , \(\sum _{j=0}^k \gamma _j \eta ^i_{j+1}\)  is a martingale with an almost surely bounded square variation (since \(\sum _{j=0}^{+\infty } \gamma _j^2 < + \infty \) ). The proof is concluded by applying [1]}.

In all the sequel, it is implicitly assumed that \(\epsilon \)  was chosen small enough to satisfy the assumption of Theorem REF .
Denote by \(k_{0, \epsilon }\)  the smallest integer after which we do not perform the clipping step in Algorithm REF .
\(k_{0, \epsilon }:= \inf \lbrace  & k \ge 0: \textrm { for } m \ge k\, , s^c_{\epsilon , \alpha }(\widehat{\nabla } \ell (w_m), w_m) =\\& s_{\epsilon , \alpha }(\widehat{\nabla } \ell (w_m), w_m)\rbrace  \, .\) 
